[5] Of great importance to electromagnetic wave calculations is the WKB approximation. This is a high frequency approximation that leads to the ray tracing techniques which lie at the heart of many propagation calculations. Consider expression (6) for the propagation kernel, then the maximum contribution to the path integral will come from paths around that for which the phase distance S is minimum (the ray path of Fermat's principle). Around this path, the lowest order variation in S will be quadratic. Ignoring variations higher than quadratic, this yields the WKB approximation [*Rosen*, 1969]

where **S**_{c} and δ^{2}**S**_{c} represent **S** and its second variation when evaluated on the path of minimum phase. (The remaining path integral is in terms of variations q from the path of minimum phase and for which q = 0 at the end points.) Consider the path of minimum phase to be parameterized by its group path τ (dτ = ) and denote the end points (*x*_{0}, *z*_{0}) and (*x*, *z*) by A and B, respectively. Let r = r_{c}(τ) be the minimum phase path expressed in three- dimensional coordinates, then this path satisfies [*Jones*, 1979]

Consider path variations q of the form

where the vectors P^{i} are unit length, perpendicular to the minimum phase path (and to each other) and satisfy

Vectors P^{i} are the “polarisation” vectors of the standard WKB solution to Maxwells equations [*Jones*, 1979, p. 594]. The second variation δ^{2}**S**_{c} can now be expressed in the form

where (x_{1}, x_{2}, x_{3}) are Cartesian coordinates in three-dimensional space. The normalization kernel

can be built up from the short path kernels

where Δτ is small, matrix Γ has coefficients γ^{ij} and subscripts A and B denote quantities evaluated at the beginning and end of the short path (note that the normalization in (17) ensures a delta function characteristic as Δτ 0). Consider the path of minimum phase that joins two general points and break this into N short paths, each with group path length Δτ. Repeated application of the short path kernel will yield

where the i^{th} path runs between points q_{i−1} and q_{i} ( note that q_{0} = q_{N} = 0). The above integral can be evaluated to yield

where

with

where δ^{ij} is the Kronecker delta. It will be noted that both C^{12} and *F* = *C*^{11} − *C*^{22} are diagonal matrices and that *D* = Δτ^{2}∣*C*^{22}^{2} + *C*^{22}*F* − *C*^{12}^{2}∣. From the factorization

it follows that

where

and

Expanding the above determinant results in the difference equation

and, in the limit N ∞, the differential equation

with boundary conditions 1 and **D**_{±} 0 as τ 0 (it has been assumed that *n* = 1 at the start of propagation).

[6] Equations (11), (13), (15) and (27) together provide a means of calculating the parameters that are required in the evaluation of the kernel

This kernel, together with integral equation (4), provides the required WKB solution to the Helmholtz equation (S(r, r_{0}) = is the phase distance). Although the calculation of the kernel will require both P^{1} and P^{2} , it will be noted that P^{1}, P^{2} and the minimum phase path are mutually orthogonal and so the solution of (13) will only be required for either P^{1} or P^{2}. It will also be noted that a knowledge of P^{1} and P^{2} facilitates the calculation of a WKB solution to the full Maxwell equations for nonhomogeneous media (a fact will be exploited in the next section). At the beginning of propagation, the electric field will need to be expanded in the P^{1} and P^{2} directions. The evolution of these vectors can then be used to track the evolution of the electric field direction along a ray and the above WKB kernel its magnitude. This procedure will provide an effective approximate solution away from caustics and strong sources of diffraction.